A new Japanese prime minister is likely to come into the international limelight before the end of the year. He will face the same tiger at the front door that the present prime minister, Yasuhiro Nakasone, grapples with - his country's huge trade surplus with the rest of the world. The wolf at the back door, unemployment, will be less familiar.

That is the prediction Japanese economic analysts are making, as the Year of the Tiger gives way to that of the rabbit, according to the Japanese calendar.

Mr. Nakasone has said that he is going to spend a lot of time listening, rabbit-like, to world currents as well as to domestic ones. And when he moves, he says, it will be with a rabbit's speed.

His term expires at the end of October, and almost no one expects the ruling Liberal Democrats to give him an extension. After nearly five years in office by then, he will be Japan's third-longest-ruling premier.

Who will succeed him? Finance Minister Kiichi Miyazawa, an internationalist who is fluent in English? Shintaro Abe, who visited 81 countries during four years as foreign minister, ending last summer? Or Noboru Takeshita, wily secretary-general of the Liberal Democratic Party?

Mr. Abe is much better known internationally, but Mr. Takeshita, a politician's politician, is considered to have the inside track if only he can unite all 140 members of his Tanaka faction behind him. Takeshita's faction, the largest and most powerful of the five major Liberal Democratic factions, is so called because it was long led by former Prime Minister Kakuei Tanaka.

The most thoughtful of the trio, Mr. Miyazawa, sits in the hot seat as finance minister, and his chances will depend largely on how successfully he manages to shepherd the budget through the Diet (parliament) and then to get the government's tax-reform bill enacted.

Besides these domestic preoccupations, there is that tiger at the front gate - a trade surplus with the world, but more particularly with the United States. Japan's surplus with the US in 1985 was nearly $40 billion. In 1986 it is likely to be closer to $50 billion, despite a 40 percent rise in the value of the yen, which is gradually reducing Japan's exports in terms of volume and of yen valuation.

Now that Democrats control both houses of the United States Congress, an omnibus trade bill enacting stiff penalties against Japan and other countries with chronic trade surpluses is likely to override a presidential veto.

Today the yen stands at less than 160 to the dollar, compared with nearly 240 to the dollar in September 1985. If, out of exasperation with the continuing flow of Japanese cars, consumer electronics, and computer chips into the American market, Washington allows the dollar to slide even further against the yen, some of yesteryear's most of Japan's basic industries - coal, nonferrous metals, shipbuilding, and steel - will face retrenchment problems even more traumatic than the ones that bedevil them today.

Unemployment, the wolf at Japan's back door, is increasing. Except in a few declining industries such as coal or textiles, it has been practically unknown during all the years of Japan's rapid economic growth. Even now, with a mature economy, it stands at only 2.8 percent, still low by US or West European standards. But certain think tanks here are predicting a rise to 3.3 percent by the year's end, or about 2 million people.

For many Japanese, this is a frightening figure. The social stability that has been one of the underpinnings of Japan's phenomenal economic growth is bound to be affected.

And then there is tax reform - popular in some business circles, but controversial with the public. Income tax is to be reduced somewhat, but the main new tax the public sees is a 5 percent value-added tax that seems to violate the Liberal Democrats' campaign pledge last summer to introduce no large new taxes. Tax reform also eliminates long-sacrosanct tax exemptions on small savings (up to about $18,000 per person).

The Liberal Democrats have a comfortable majority of 305 in Japan's House of Representatives, but the opposition parties will use every parliamentary maneuver to oppose the proposal and delay the voting, and the conclusion may not come before late summer.

None of Nakasone's prospective successors wants to be the one to have to tackle tax reform in his administration. Nor have any of them come up with alternative programs likely to win greater acceptance from the Japanese public. In the end, therefore, the Liberal-Democrats are expected to push tax reform through the Diet, using their parliamentary majority.

Some politicians even contemplate a kind of unspoken deal with the opposition whereby tax reform will become law but the Nakasone Cabinet would be forced to resign. That would be a less-than-honorable end for a prime minister, one of whose greatest achievements has been to give Japan a higher profile in the councils of the world, and especially in relations with the Reagan White House.

If Nakasone had his druthers, he would probably like to be host to Soviet leader Mikhail Gorbachev's first visit to Japan and go out in a blaze of glory before the memories of that event fade. So complicated are the twists and turns of domestic politics, however, and so intense is the combination of external and internal pressures being brought to bear on Japan, that the Year of the Rabbit may turn out to be one of the most unpredictable 12 months this country has experienced since its postwar recovery began.

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